Issue 
Mechanics & Industry
Volume 21, Number 3, 2020



Article Number  305  
Number of page(s)  16  
DOI  https://doi.org/10.1051/meca/2020017  
Published online  03 April 2020 
Regular Article
A multiobjective optimization of journal bearing with double parabolic profiles and groove textures under steady operating conditions
^{1}
College of Power and Energy Engineering, Harbin Engineering University, Harbin, PR China
^{2}
Chongqing Yuejin Machinery Plant Co., Ltd, Chongqing, PR China
^{*} email: donglizhaobin@hrbeu.edu.cn
Received:
27
December
2019
Accepted:
20
February
2020
The double parabolic profiles can help journal bearing to reduce bushing edge wear, but it also reduces load carrying capacity and increases friction loss. To overcome these drawbacks, in this study, a multiobjective optimization of journal bearing with double parabolic profiles and groove textures is researched under steady operating conditions using Taguchi and grey relational analysis methods. Firstly, a lubrication model with journal misalignment, elastic deformation, asperity contact, thermal effect is established and formation cause of drawbacks is illustrated. Then, an orthogonal test with considering six factors, i.e., groove number, groove depth, groove length, axial width of double parabolic profiles, radial height of double parabolic profiles and groove location is conducted, meanwhile the effects and significances of each factor on response variables are revealed. Finally, an optimal parameters combination of six factors is determined by grey relational analysis, which gives maximum load carrying capacity and minimum friction loss. Overall, this study may give guidance on journal bearing design to enhance its tribological performance.
Key words: Double parabolic profiles / Groove textures / Load carrying capacity / Friction loss / Multiobjective optimization
© AFM, EDP Sciences 2020
1 Introduction
Journal bearing is a critical component in practical engineering. As the journal misalignment, deformation, machining and installation errors are unavoidable, bushing edge wear is found in some applications, as illustrated in Figure 1. Earlier study [1] had shown the double parabolic profiles can reduce bushing edge wear, but it also reduces load carrying capacity and increases friction loss, which still need to be well solved.
For mechanical components, surface texturing has been widely used in the past decades to improve their contact performance [2]. Specially, effects of surface textures on performance of journal bearing also attracted wide attentions to scholars. Ji et al. [3] employed sinusoidal waves to characterize rough surface, which showed the greater roughness ratio can suppress the hydrodynamic effect of textures significantly. Hence, it is necessary to minimize roughness of textured surface. Gu et al. [4] presented a mixed lubrication model to analyze the performance of groove textured journal bearing with nonNewtonian fluid operating from mixed to hydrodynamic lubrications. Their results showed the surface texturing can increase the asperity contacts, but the contact behaviors mainly arise in first cycle of startup process. When journal bearing works under normal operating conditions, the wear due to asperity contacts will be small. Literatures [5–10] showed the partial textures can positively affect bearing performance, but the optimal locations are some different depending on geometrical parameters and working conditions [5]. Yu [6] and Lin [7] showed the textures located at rising phase of pressure field increases load carrying capacity, while the textures located at falling phase of pressure field reduces load carrying capacity. However, TalaIghil et al. [8,9] showed the textures located in declining part of pressure field can generate extra hydrodynamic lift. Shinde and Pawar [10] pointed out among three partial grooving configurations (90–180^{0}, 90–270^{0}, 90–360^{0}), the first configuration gives the maximum pressure increase while the last configuration gives the minimum frictional power loss. Their results indicate the optimal location also depends on the optimization target.
Studies mentioned above have shown the texture performance is affected by many factors, which makes it very complicated to obtain the optimal design. To address this issue, some optimization methods are adopted by researchers, including the genetic algorithm [11–13], neural network [14] and sequential quadratic programming [15], while these methods are somewhat difficult in mathematics. By contrast, Taguchi method may provide a handy way for this issue. Taguchi assumes that introducing quality concepts at design stage may be more valuable than inspection after manufacturing. Hence, Taguchi method aims to optimize process to minimize quality loss with objective functions such as “thenominalthebest”, “thelargerthebetter”, or “thesmallerthebetter” depending on experimental objectives [16]. In practical engineering, this method uses the concept of orthogonal array to reduce the numbers of experiments, which facilitates to research multiparameters concurrently and evaluate the effects of each parameter. Some studies [17–19] have already adopted Taguchi method to optimize surface textures for journal bearing to maximize load carrying capacity, minimize side leakage and friction loss.
Despite of remarkable progress on research of surface textures, few studies have been researched the journal bearing with double parabolic profiles and groove textures. The novelty of this study is to adopt the Taguchi and grey relational analysis methods to conduct a multiobjective optimization of journal bearing with double parabolic profiles and groove textures, i.e., double parabolic profiles are applied to eliminate bushing edge wear, while groove textures are applied to overcome the negative effects of the former, such as reduced load carrying capacity and increased friction loss. The results show this study may help journal bearing to enhance its tribological performance.
Fig. 1 The bushing edge wear. 
2 Lubrication model
2.1 Geometric model
Figure 2 illustrates the layout of double parabolic profiles and groove textures investigated in this study. In Figure 2, B and d_{b} are the bushing width and thickness; L_{y } and L_{z } are the axial width and radial height of double parabolic profiles, whose equation can be described as δ_{z } = (L_{z }/L_{y } ^{2}) y ^{2}; (θ_{s } – θ_{e }), w_{g}, d_{g}, l_{g} and w_{e}, are the groove location, width, depth, length and gap, and their specific values will be introduced in Section 4.
Fig. 2 Layout of double parabolic profiles and groove textures. 
2.2 Film thickness
Figure 3 illustrates a misaligned journal bearing under external moment M_{e} , whose lubricating oil is supplied through the axial oil feeding groove. For simplicity, only the misalignment in vertical plane yoz is considered.
As the elastic module of journal is much higher than that of bushing, only elastic deformation of bushing surface is considered. Thus, the nominal film thickness h is(1)where h_{g} is the nominal film thickness without elastic deformation, which is(2)where c is the radial clearance, e the eccentricity of the midplane, ϕ the attitude angle of the midplane, y the axial coordinate, γ the misalignment angle, δ_{z } the clearance added by double parabolic profiles, δ_{tex } the clearance added by groove textures. Obviously, for journal bearing with plain profile, δ_{z } = δ_{tex } = 0, and for journal bearing with only double parabolic profiles, δ_{tex } = 0.
In this study, the elastic deformation δ_{e } is obtained by Winkler/Column model [20], which gives a simpler way than finite element method [21] to estimate elastic deformation and has been used in literatures [22–24]. The model assumes the local elastic deformation is only dependent on local film pressure, as expressed in equation (3) (3)where ν is the Poisson's ratio of alloy layer, E the elastic modulus of alloy layer, p the film pressure, d the thickness of alloy layer.
Fig. 3 A misaligned journal bearing. 
2.3 Reynolds equation
The Reynolds equation based on average flow model is utilized to determine the roughness effects on performance of journal bearing [25,26], as expressed in equation (4) (4) where μ is the viscosity of lubricating oil, p the film pressure, U_{1} and U_{2} the velocities of two surfaces, σ the standard deviation of combined roughness, φ_{x }, φ_{y } the pressure flow factors, φ_{s } the shear flow factor, h_{T} the local film thickness.
For journal bearing under steady operating conditions, equation (4) can be expressed as followed by the variable transformation x = Rθ (5)where ω is angular velocity of journal.
2.4 Asperity contact pressure
The asperity contact model proposed by Greenwood and Tripp [27] is utilized here to estimate interaction effects of asperities, which is widely used in the analysis of rough surfaces contact of journal bearing. The asperity contact pressure P_{asp} is given by(6)where η is the number of asperities per unit area, β the mean radius of curvature of the asperities, σ the standard derivation, E_{c} the composite elastic modulus, F_{2.5} (h/σ) the Gaussian distribution function. Note the surface pattern parameter γ is assumed as 1, which means the roughness structures are isotropic.
2.5 Friction loss
It is assumed when journal bearing operates in mixed lubrication, the total friction force consists of hydrodynamic friction force arising from the shearing of lubricating oil and asperity contact friction force [28]. Hence, the total friction force f is(7)where U = ω R, φ_{f }, φ_{fs }, φ_{fp } are shear stress factors, μ_{asp} boundary friction coefficient, here μ_{asp } = 0.02. The friction loss P_{f} can be calculated by(8)
2.6 Leakage flowrate
The leakage flowrate Q_{1} from frontend plane of bearing and Q_{2} from rear end plane of bearings are [21](9)
The total leakage flowrate Q is(10)
2.7 Thermal effect
As well known, the temperature of lubricating oil will increase and its viscosity will decrease during operations, so it is more accurate to adopt a variable viscosity model in calculation. In this study, an effective temperature is obtained based on the adiabatic flow hypothesis of lubricating oil, as shown below(11)where T_{e} is the effective temperature of lubricating oil, T_{i} the inlet oil temperature, P_{f} the friction loss, Q the total leakage flowrate, ρ the density of lubricating oil, c_{l} the specific heat of lubricating oil, k the correction factor and k = 0.9 [29]. This simple method avoids the complex computation of thermohydrodynamic lubrication and has been used in the literatures [24].
CD40 lubricating oil is used here and its viscositytemperature equation can be expressed as(12)where the unit of μ is Pa·s, and a = 6.163, b = 8.721 × 10^{−5},c = − 0.0455, respectively. Once the effective temperature is obtained, the effective viscosity can be calculated by the equation (12).
2.8 Load equilibrium
In this study, the external load is assumed as a pure moment whose direction is parallel to x axis, which only leads to a journal misalignment in vertical plane yoz. The static equilibrium of journal center can be described as(13)where M_{e} is the external moment, M_{t} the resultant moment of hydrodynamic moment M_{oil} and asperity contact moment M_{asp} , namely M_{t} = M_{oil} + M_{asp} .
The load equilibrium equations along x and z axis are(14)where M_{ex} and M_{ez} are the external moment along x and z axis, M_{tx} and M_{tz} the resultant moment along x and z axis, which can be expressed as follows(15) where M_{oilx} and M_{oilz} are the hydrodynamic moment along x and z axis, M_{aspx} and M_{aspz} the asperity contact moment along x and z axis, which can be calculated by(16) (17)
3 Numerical procedure and verification
Apply the finite difference method to discretize equation (5), then solve the difference equations by overrelaxation iterative method. Reynolds boundary conditions are used to determine the rupture zone of oil film, and the pressures in oil feeding groove and both bearing ends are assumed as zero. The discretized pressure can be calculated by(18)where p_{i,j } ^{(kp + 1) } is the film pressure for node (i, j) at the (k_{p}+1)th iteration, p_{i,j } ^{(kp ) } the film pressure for node (i, j) at the k_{p}th iteration, ω_{s} the overrelaxation factor, here ω_{s} =1.5. DD_{i,j}, CS_{i,j}, CN_{i,j}, CE_{i,j}, CW_{i,j}, CC_{i,j} are the difference coefficients during the pressure solution.
The film pressure convergence criteria at the k_{p}th iteration is given by(19)where ϵ_{p } is the allowable precision of pressure solution, here ϵ_{p } = 10^{−5}. n_{θ } and n_{y } are the nodes numbers along circumferential and axial direction.
Based on the equation (11), the effective temperature convergence criteria at the k_{t}th iteration is given by(20)where ϵ_{t } is the allowable precision of effective temperature solution, here ϵ_{t } = 10^{−4}.
When journal bearing operates in steady operations, it can be assumed the sum of hydrodynamic and asperity contact moments is approximately equal to the external moment, specifically, the motion of journal can be obtained by correcting eccentricity ratio ϵ (ϵ = e/c), attitude angle ϕ and misalignment angle γ. The correction strategies can be expressed as follows(21) (22) (23)where M_{e} is the amplitude of M_{e} , M_{t} is the amplitude of M_{t} , ω_{ϕ }, ω_{ϵ }, ω_{γ } are the correction factors of ϕ, ϵ, γ, and ω_{ϕ } = 0.9, ω_{ϵ } = 10^{−2}, ω_{γ } = 10^{−5}, respectively.
As mentioned above, the external load is a pure moment whose direction is parallel to the x axis. Accordingly, the equilibrium convergence criteria can be given by(24) (25)where err_{xz} and err_{M} are allowable precisions for the equilibrium calculation, here err_{xz } = 10^{−3}, err_{M } = 10^{−3}.
The whole computational process is shown in Figure 4, and the programing platform is Intel Core i74790 at 3.6GHz with 32GB RAM.
Before the following analysis, it is necessary to validate the model. The calculated pressures are compared with Ferron's experimental results [30], as illustrated in Figure 5. The comparisons show the calculated results agree well with the experimental ones, which confirms the validity of this model.
Fig. 4 Flow chart of the computational process. 
4 Results and discussions
The detailed parameters of the journal bearing investigated in this study are listed in Table 1.
Mesh refinement analysis is conducted based on the journal bearing with plain profile. Various mesh schemes and corresponding minimum film thickness (h_{min}) are listed in Table 2. It can be seen that h_{min} is converged when the mesh is 1441 × 181. Considering the solving time and accuracy, 1441 × 181 mesh is adopted.
Here one typical case is conducted to prove the bushing edge wear can be reduced by double parabolic profiles. For journal bearing with double parabolic profiles, the chosen values of axial width L_{y } and radial height L_{z } are 10 mm and 6 μm, and the relative variation of each performance parameter is defined as(26)where δ_{dpp } X is the relative variation of parameter X, pp means plain profile and dpp means double parabolic profiles. Note in this study, the maximum film pressure (P_{max}) is used to indirectly reflect the variations of load carrying capacity, smaller P_{max} means better load carrying capacity and vice versa.
The comparison results are listed in Table 3. It can be observed that although the ϵ and γ both increase to some extent, the h_{min} has sharply increased and the P_{aspmax} has reduced to zero in dpp case, which confirms its validity in term of reducing edge wear. Note ϕ has reduced a little, this variation trend indicates the bearing stability is also improved as the dpp reduces effective bearing width, which agrees with the common sense that using short bearing is favorable for stability improvement. However, the load carrying capacity is reduced as the dpp leads to a less capacity area. Hence, greater P_{max} is generated and hydrodynamic friction force also increases which causes greater P_{f}. It is obvious that Q is increased as the dpp increases the film pressure gradient at bushing edge, so the comprehensive effects of greater P_{f} and Q lead to a tiny change of T_{e} based on equation (11).
Figure 6 illustrates the variations of film thickness and pressure between pp and dpp case, in Figure 6a, Δh = h_{dpp } − h_{pp }, and in Figure 6b, ΔP = P_{dpp } − P_{pp }. It can be seen that, compared with pp case, the dpp case presents thicker oil film at bushing edge, which avoids the asperity contacts between journal and bushing surfaces, so the asperity contact pressure reduces to zero. However, in the region away from bushing edge, the oil film in dpp case is thinner than that in pp case, which causes greater film pressure here, i.e., the dpp can move the film pressure peak to inside region.
As mentioned above, although dpp can reduce bushing edge wear, it also reduces load carrying capacity and increases friction loss. The main motivation of this study is to overcome these drawbacks by groove textures, as illustrated in Figure 2. Considering the journal bearing with double parabolic profiles and groove textures (dppgt), six factors are involved in the analysis: groove number n_{g}, depth d_{g}, length l_{g}, axial width of dpp L_{y }, radial height of dpp L_{z }, and location (θ_{s } – θ_{e }). Three levels for each factor are listed in Table 4. Note the total area of groove textures, S_{g} = w_{g} × l_{g} × n_{g}, is constant, while the groove width w_{g}, gap w_{e}, and number n_{g} are different. The values of w_{g} and w_{e} for corresponding n_{g} are listed in Table 5.
Clearly the study of six factors at three optional levels needs 3 [6] simulation tests if full factorial designs are implemented. To reduce the computing cost, Taguchi method with orthogonal array L_{18} is adopted here, as shown in Table 6. L_{18} is commonly used and it is more concise than L_{27} array.
As we are mainly concerned with load carrying capacity and friction loss, only the results of P_{max} and P_{f} are given in Table 7. Note reference group are the results of dpp case (L_{y } = 5, 10, 15 mm, L_{z } = 3, 6, 9 μm, totally 9 cases, each case will be repeated twice).
As can be seen from Table 7, compared with reference cases, the groove textures show positive effects only in No. 3, 4, 8, 11, 13 and 18 tests (shown in bold). Take the No. 8 test to explain this phenomenon: As illustrated in Figures 7 and 8, the groove textures can strengthen the hydrodynamic effect of lubricating oil and generate extra local pressure at the downstream of film pressure filed, which increases the load carrying capacity. Meanwhile, the friction force arising from the shearing of oil reduces with thicker oil film, which causes less friction loss. Note the groove locations of these six tests are all 200°–350°, which indicates the proper groove location is beneficial to performance enhancement.
While for the remaining tests, the groove textures show negative effects on either load carrying capacity or friction loss compared with reference cases. Take No. 5 and 15 tests to explain this phenomenon: As illustrated in Figures 9–12, the groove textures located at 180°–330° (No. 5 test) and 190°–340° (No. 15 test) increase film thickness at main loading region, so the continuous film pressure generation is destroyed and greater pressure is generated in multipeaks at untextured land (region between two adjacent grooves). This phenomenon is more evident when the groove textures locates at 180°–330°. Meanwhile, the hydrodynamic friction loss also reduces as explained previously. Moreover, the friction loss in No. 1, 9, 12, and 16 orthogonal tests are greater than those in reference cases, coincidentally the groove locations of these four tests are all 180°–330°, which indicates the improper groove location may bring harmful effects on texture performance.
Based on the orthogonal tests, the main effect analysis and analysis of variance (ANOVA) are performed to show the effects and significance of each factor [31]. The effects of six factors on load carrying capacity (P_{max}) are illustrated in Figure 13. It is observed the optimal parameters combination is at groove number 120, depth 10 μm, length 50 mm, axial width of dpp 10 mm, radial height of dpp 3 μm, and location 200°–350°, which may give a best load carrying capacity. However, this combination does not exist in orthogonal table and another computation is conducted, denoted as No. 19 test. The results of this test are P_{max} = 64.0170 MPa and P_{f} = 4105.1540 W, respectively, and P_{max} of all 19 tests are illustrated in Figure 14. It can be seen the No. 8 test, i.e., groove number 30, depth 15 μm, length 60 mm, axial width of dpp 10 mm, radial height of dpp 3 μm, and location 200°–350°, gives a maximum load carrying capacity, with only 4.69% increases of P_{max} than that in pp case.
Table 8 lists the results of ANOVA for load carrying capacity (P_{max}). The columns represent the sources, degrees of freedom (DF), sum of squares (SS), mean of squares (MS), Fvalues, and F_{0.05} (2, 5). Table 8 shows the groove number n_{g}, location θ_{s } – θ_{e } and depth d_{g} are the significant factors at 95% confidence level as their Fvalues are greater than F_{0.05} (2, 5). The percentage contributions (PCR) of all factors are also given in Table 8, which shows the groove number n_{g} is the most important factor whose percentage contribution is 38.66%, followed by location θ_{s } – θ_{e } and depth d_{g}, whose percentage contribution are 25.78% and 16.85%.
The effects of six factors on the friction loss (P_{f}) are illustrated in Figure 15. It is clearly that Figure 15 suggests the optimal parameters combination is in the No. 6 test, i.e., groove number 60, depth 20 μm, length 60 mm, axial width of dpp 5 mm, radial height of dpp 3 μm, and location 190^{°}–340^{°}, which gives a minimum friction loss. The P_{f} of all 19 tests are illustrated in Figure 16, which also confirms the No. 6 test gives a minimum friction loss, 3643.9945 W, with 11.78% decrease than that in pp case.
Table 9 lists the results of ANOVA for friction loss (P_{f}), which shows the groove number n_{g}, depth d_{g}, axial width of dpp L_{y}, radial height of dpp L_{z} and location θ_{s } – θ_{e } are the significant factors at 95% confidence level. The percentage contributions of all factors are also given in Table 9, which shows the groove number n_{g} is the most important factor whose percentage contribution is 33.45%, followed by the location θ_{s } – θ_{e }, depth d_{g}, axial width of dpp L_{y} and radial height of dpp L_{z} whose percentage contributions are 21.57%, 21.00%, 11.39% and 8.04% respectively.
Note the interactions between factors are not considered in. In fact, the interactions can be neglected if the orthogonal table is reasonably designed. The ANOVA shows that, compared with the six factors, the PCR of errors are small, 6.62% of P_{max} and 2.70% of P_{f}, which indicates the interactions hidden in error are very limited. It can be found the interactions are also neglected in literatures [17–19].
From the above analysis, it can be observed the No. 8 test gives a maximum load carrying capacity while the No. 6 test gives a minimum friction loss. To find an optimal configuration, the grey relational analysis (GRA) method is used for this multiobjective optimization [19]. The main steps of GRA are listed as follows:
Step 1: Solution for normalized sequence.
As the optimization objectives are increasing load carrying capacity (smaller P_{max}) and reducing friction loss (smaller P_{f}), the “thesmallerthebetter” criterion is used to normalize the orthogonal test results between 0 and 1, as shown below(27)where the X_{i} ^{*} (k) is the normalized sequence, x_{i}(k) the sequence of P_{max} or P_{f}, k =1, 2 (1 for P_{max} and 2 for P_{f}), and i = 1, 2,…, 19 (test No.).
Step 2: Solution for deviation sequence.
The deviation sequence, denoted as Δ _{0i } (k), is the absolute difference between the reference and normalized sequences, as shown below(28)where X_{0} ^{*}(k) is the reference sequence.
Step 3: Solution for grey relational coefficient (GRC).
The GRC is calculated depending on the deviation sequence to describe the correlation between the reference and normalized sequences, as shown below(29)where Δ _{min} and Δ _{max} are the minimum and maximum values of Δ _{0i } (k), ζ the distinguishing factor between 0 and 1, here ζ = 0.5, which means the increasing load carrying capacity and reducing friction loss are equally important.
Step 4: Solution for grey relational grade (GRG).
The evaluation criteria for a multiobjective optimization is based on GRG, which is the mean of the GRCs, as shown in equation (30). The larger GRG means the parameter configuration approaches the optimal one.(30)Based on above four steps, the GRGs of all 19 tests are calculated and illustrated in Figure 17. It is observed the GRG of No. 8 test is the largest one, which means when both considering the load carrying capacity and friction loss, the parameters of No. 8 test, i.e., groove number 30, depth 15 μm, length 60 mm, axial width of dpp 10 mm, radial height of dpp 3 μm, and location 200^{°}–350^{°}, is the optimal parameters combination. In this case, the P_{max} is 62.8141 MPa and P_{f} is 4113.0399 W, with only 4.69% increase while 0.42% decrease than those in the pp case, and meanwhile the bushing edge wear is totally eliminated.
At last, it should be noted that the Taguchi and grey relational analysis methods gives the optimal parameters combination only limited to the preassigned parameter levels, which may miss the really optimal design. However, they are relatively simple and practicable, which facilitates the research of multiparameters and evaluation of parameter effects.
Detailed parameters of the journal bearing.
Mesh schemes for mesh refinement analysis.
Comparisons between journal bearing with plain profile (pp) and double parabolic profiles (dpp).
Fig. 6 (a) Differences Δh, (b) Differences ΔP between pp and dpp cases. 
Six control factors and their optional levels.
Groove sizes for their optional levels.
Orthogonal array L18.
Results of orthogonal tests and reference group.
Fig. 7 (a) Film thickness of No. 8 test, (b) Film thickness of reference case 8. 
Fig. 8 (a) Film pressure of No. 8 test, (b) Film pressure of reference case 8. 
Fig. 9 (a) Film thickness of No. 5 test, (b) Film thickness of reference case 5. 
Fig. 10 (a) Film pressure of No. 5 test, (b) Film pressure of reference case 5. 
Fig. 11 (a) Film thickness of No. 15 test, (b) Film thickness of reference case 15. 
Fig. 12 (a) Film pressure of No. 15 test, (b) Film pressure of reference case 15. 
Fig. 13 Main effect plots of P_{max} (MPa). 
Fig. 14 P_{max} of all 19 tests. 
ANOVA for P_{max}.
Fig. 15 Main effect plots of P_{f} (W). 
Fig. 16 P_{f} of all 19 tests. 
ANOVA for P_{f}.
Fig. 17 GRGs of all 19 tests. 
5 Conclusions
In this study, based on Taguchi and GRA methods, a multiobjective optimization of journal bearing with double parabolic profiles and groove textures is researched under steady operating conditions. Following conclusions can be drawn from numerical results:

Compared with plain profile, double parabolic profiles can eliminate bushing edge wear, and the stability of journal bearing can be also improved. However, it also has visible drawbacks, such as the reduced load carrying capacity (with 13.51% increase of P_{max}) and increased friction loss (with 3.46% increase of P_{f}).

Compared with double parabolic profiles, double parabolic profiles with proper groove textures can increase the load carrying capacity due to extra local pressure. Meanwhile, the friction loss also reduces with thicker oil film. The improper groove textures can deteriorate the performance as they will destroy continuous film pressure generation and increase friction loss.

The main effects analysis based on orthogonal test results shows that, for load carrying capacity, the No.8 test gives a minimum P_{max}, 62.8481 MPa, with only 4.69% increase than that in the case of plain profile. While for friction loss, the No. 6 test gives a minimum P_{f}, 3643.9945 W, with 11.78% decrease than that in the case of plain profile.

The ANOVA shows that, for load carrying capacity, the groove number n_{g} and groove location θ_{s } – θ_{e } are the significant factors at 95% confidence level, whose percentage contributions are 38.66%, 25.78%, 16.85% respectively. While for friction loss, the groove number n_{g}, groove location θ_{s } – θ_{e }, groove depth d_{g}, axial width of double parabolic profiles L_{y} and radial height of double parabolic profiles L_{z} are the significant factors at 95% confidence level, whose percentage contributions are 33.45%, 21.57%, 21.00%, 11.39%, and 8.04% respectively.

The GRA shows that, the parameters combination of No. 8 test, i.e., groove number 30, groove depth 15 μm, groove length 60 mm, axial width of double parabolic profiles 10 mm, radial height of double parabolic profiles 3 μm, and groove location 200^{°}–350^{°}, is the optimal solution: P_{max} is 62.8141 MPa and P_{f} is 4113.0399 W, with only 4.69% increase while 0.42% decrease than those in the case of plain profile, meanwhile the bushing edge wear is eliminated.
The cavitation area of oil film determined by Reynolds boundary conditions may be less accurate than massconservative treatment, which is a limitation of this study. In future work, cavitation effects will be treated in massconserving way, and other optimization methods will also be tried for journal bearing to find the optimal texture design.
Nomenclature
d : Thickness of plain profile
L_{y} : Axial width of double parabolic profiles
L_{z} : Radial height of double parabolic profiles
h_{g} : Nominal film thickness without elastic deformation
e : Eccentricity of the midplane
ϕ : Attitude angle of the midplane
δ_{z } : Variation clearance caused by double parabolic profiles
δ_{tex } : Variation clearance caused by groove textures
δ_{e} : Elastic deformation of bushing surface
E : Elastic modulus of the bushing
ν : Poisson's ratio of the bushing
μ : Viscosity of lubricating oil
U_{1}, U_{2} : Velocities of the two surfaces
σ : Standard deviation of combined roughness
ϕ_{x},ϕ_{y} : Pressure flow factors
ω : Angular velocity of journal
P_{asp} : Asperity contact pressure
η : The number of asperities per unit area
β : The mean radius of curvature of the asperities
F_{2.5} (h/σ): Gaussian distribution function
ϕ_{f},ϕ_{fs},ϕ_{fp} : Shear stress factors
μ_{asp} : Boundary friction coefficient
Q_{1} : Leakage flowrate from the front end plane
Q_{2} : Leakage flowrate from the rear end plane
T_{e} : Effective temperature of lubricating oil
ρ : Density of lubricating oil
c_{l} : Specific heat of lubricating oil
M_{asp} : Asperity contact moment
M_{ex}, M_{ez} : External applied moment along the x and z axes
M_{tx}, M_{tz} : Resultant moment along the x and z axes
M_{oilx}, M_{oilz} : Hydrodynamic moment along the x and z axes
M_{aspx}, M_{aspz} : Asperity contact moment along the x and z axes
ϵ_{p} : Allowable precision for the solution of film pressure
ϵ_{t} : Allowable precision for the solution of the effective temperature
ϵ : Eccentricity ratio of the midplane
M_{e} : Amplitude of external moment
M_{t} : Amplitude of resultant moment
ω_{ϕ } : Correction factor of ϕ
ω_{ϵ } : Correction factor of ϵ
ω_{γ } : Correction factor of γ
err_{xz}, err_{M} : Allowable precision for the calculation of load equilibrium
σ_{b} : Standard deviations of the roughness of the bearing surface
σ_{j} : Standard deviations of the roughness of the journal surface
n_{θ }, n_{y} : Numbers of nodes along the circumferential and axial direction
h_{min} : Minimum film thickness
P_{max} : Maximum film pressure
P_{asp max} : Maximum asperity contact pressure
n_{gθ } × n_{gy} : Mesh of single groove
S_{g} : Total area of groove textures
X_{i} ^{*} (k): Normalized sequence of P_{max} or P_{f}
x_{i}(k): Sequence of P_{max} or P_{f}
X_{0} ^{*}(k): Reference sequence
Δ _{0i } (k): Deviation sequence
Δ_{min}, Δ_{max} : Minimum and maximum values of Δ _{0i } (k)
ζ : Weight of P_{max} or P_{f}
ξ_{i} (k): Grey relational coefficient
Abbreviations
dpp: Double parabolic profiles
dppgt: Double parabolic profiles with groove textures
GRC: Grey relational coefficient
Acknowledgments
This work is supported by the National Natural Science Foundation of China (51809057) and Fundamental Research Funds for the Central Universities (3072019CFM0302).
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Cite this article as: C. Liu, N. Zhong, X. Lu, B. Zhao, A multiobjective optimization of journal bearing with double parabolic profiles and groove textures under steady operating conditions, Mechanics & Industry 21, 305 (2020)
All Tables
Comparisons between journal bearing with plain profile (pp) and double parabolic profiles (dpp).
All Figures
Fig. 1 The bushing edge wear. 

In the text 
Fig. 2 Layout of double parabolic profiles and groove textures. 

In the text 
Fig. 3 A misaligned journal bearing. 

In the text 
Fig. 4 Flow chart of the computational process. 

In the text 
Fig. 5 Comparisons with Ferron's experimental results [30]. 

In the text 
Fig. 6 (a) Differences Δh, (b) Differences ΔP between pp and dpp cases. 

In the text 
Fig. 7 (a) Film thickness of No. 8 test, (b) Film thickness of reference case 8. 

In the text 
Fig. 8 (a) Film pressure of No. 8 test, (b) Film pressure of reference case 8. 

In the text 
Fig. 9 (a) Film thickness of No. 5 test, (b) Film thickness of reference case 5. 

In the text 
Fig. 10 (a) Film pressure of No. 5 test, (b) Film pressure of reference case 5. 

In the text 
Fig. 11 (a) Film thickness of No. 15 test, (b) Film thickness of reference case 15. 

In the text 
Fig. 12 (a) Film pressure of No. 15 test, (b) Film pressure of reference case 15. 

In the text 
Fig. 13 Main effect plots of P_{max} (MPa). 

In the text 
Fig. 14 P_{max} of all 19 tests. 

In the text 
Fig. 15 Main effect plots of P_{f} (W). 

In the text 
Fig. 16 P_{f} of all 19 tests. 

In the text 
Fig. 17 GRGs of all 19 tests. 

In the text 
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